Figures REF (a) and REF (b) display \(\mathcal {M}_{{\rm L} \rightarrow {\rm F}}\)  for model A and model B, respectively. The plots of \(\sigma \)  (See supplemental Fig. S9) are almost indistinguishable from those of \(\mathcal {M}\) , indicating that a majority of \(\mathcal {M}\)  is actually coming from \(\sigma \) , which is due to shared history between L and F. As has been established for the Vicsek model [1]}, cohesive behavior increases as a function of density. Here,
\(\mathcal {M}_{{\rm L} \rightarrow {\rm F}}\)  and \(\sigma _{{\rm L} \rightarrow {\rm F}}\)  increase as a function of \(N_{\rm F}\)  in model A. Model B, however, is not the same as the original Vicsek model in that followers do not interact with each other, and therefore \(\mathcal {M}_{{\rm L} \rightarrow {\rm F}}\)  and \(\sigma _{{\rm L} \rightarrow {\rm F}}\)  decrease as a function of \(N_{\rm F}\) , since the inclusion of additional agents which are not interacting decreases the overall cohesion between
the present state of L and the future state of F. The plots of \(\mathcal {M}_{{\rm F} \rightarrow {\rm L}}\)  for model A and B are not shown as they
are not distinguishable by eye from those of \(\mathcal {M}_{{\rm L} \rightarrow {\rm F}}\)  (See supplemental Fig. S8).
